Property of a limit of functions of average value zero in L^2 space

  • Thread starter lmedin02
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Homework Statement



Let [itex]f_k\rightarrow f[/itex] in [itex]L^2(\Omega)[/itex] where [itex]|\Omega|[/itex] is finite. If [itex]\int_{\Omega}{f_k(x)}dx=0[/itex] for all [itex]k=1,2,3,\ldots[/itex], then [itex]\int_{\Omega}{f(x)}dx=0[/itex].

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The Attempt at a Solution


I started by playing around with Holder's inequality and constructing examples where this is the case. Since every other example I create usually does not converge to a function. I used functions defined on a bounded symmetric interval like [itex]\dfrac{1}{k}x[/itex], [itex]\sin(\dfrac{1}{k}x)[/itex]. However, I do not see how to actually set up to make this conclusion.
 

Answers and Replies

  • #2
jbunniii
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What if you write
$$\begin{align} \left| \int_{\Omega} f(x) dx \right| &=
\left| \int_{\Omega} f(x) dx - \int_{\Omega} f_k(x) dx \right| \\
&= \left|\int_{\Omega} (f(x) - f_k(x)) dx \right| \\
\end{align}$$
and apply an appropriate inequality to the right hand side?
 
  • #3
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Got it. Thanks. Using Holder's inequality and noting that the size of [itex]\Omega[/itex] is finite the results follows since [itex]f_k\rightarrow f[/itex] in [itex]L^2(\Omega)[/itex].
 
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